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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"20abfc8e272705e1e10150c5abc6b14356f4fa8393effbeff1ceed24f8a3eeaf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fefc87d068bda822","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0,               2;\n  2: 0,               2,               3;\n  3: 0,              16,              27,              4;\n  4: 0,             432,             753,            144,             5;\n  5: 0,           27648,           48840,          10220,           500,           6;\n  6: 0,         3456000,         6146472,        1342620,         75275,        1350,         7;\n  7: 0,       746496000,      1332821952,      298201216,      17937675,      381930,      3087,       8;\n  8: 0,    256048128000,    458277673536,   104060113024,    6525374045,   152527200,   1504426,    6272,     9;\n  9: 0, 131096641536000, 235022241042432, 53889814766336, 3471066652320, 85924375254, 948214512, 4930608, 11664, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(2*n^3-12*n^2+24*n-16)*x+x^2)*P(n-2,x) + (2*n^3-9*n^2+15*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*P[n - 2, x] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*v[i - 2] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-fefc87d068bda822","steps":["\u03b8+1"],"edges":["e-5d48835916960d74"],"text":"A269947 \u2192 \u03b8+1"},"headline":"A269947 \u2192 \u03b8+1","chain":null}